cp's OEIS Frontend

A370380 Array read by downward antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

%I A370380 #27 Nov 27 2024 10:05:22
%S A370380 1,1,3,1,5,13,1,7,29,71,1,9,51,195,461,1,11,79,409,1493,3447,1,13,113,
%T A370380 737,3623,12823,29093,1,15,153,1203,7427,35285,122125,273343,1,17,199,
%U A370380 1831,13601,81009,375591,1277991,2829325,1,19,251,2645,22961,164371,954419,4344485,14584789,31998903
%N A370380 Array read by downward antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.
%F A370380 Conjecture: A(n,0) = A003319(n+2). - _Mikhail Kurkov_, Oct 27 2024
%F A370380 A(n,k) = A(n,k-1) - k*A(n-1,k) + (k+2)*A(n-1,k+1) with A(n,0) = A(n-1,0) + 2*A(n-1,1), A(0,k) = 1. - _Mikhail Kurkov_, Nov 23 2024
%e A370380 Array begins:
%e A370380 ===========================================================
%e A370380 n\k|     0      1      2      3       4       5       6 ...
%e A370380 ---+-------------------------------------------------------
%e A370380 0  |     1      1      1      1       1       1       1 ...
%e A370380 1  |     3      5      7      9      11      13      15 ...
%e A370380 2  |    13     29     51     79     113     153     199 ...
%e A370380 3  |    71    195    409    737    1203    1831    2645 ...
%e A370380 4  |   461   1493   3623   7427   13601   22961   36443 ...
%e A370380 5  |  3447  12823  35285  81009  164371  304667  526833 ...
%e A370380 6  | 29093 122125 375591 954419 2124937 4289433 8025755 ...
%e A370380   ...
%o A370380 (PARI)
%o A370380 A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
%o A370380 for(i=1, m, v=vector(#v-1, k, (k+1)*v[k+1] + sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)}
%o A370380 { A(6) }
%Y A370380 Row 2 appears to be essentially A144391. - _Joerg Arndt_, Feb 17 2024
%Y A370380 Cf. A003319.
%K A370380 nonn,tabl
%O A370380 0,3
%A A370380 _Mikhail Kurkov_, Feb 17 2024